Analysis I

Code:

M1039

Acronym:

M1039

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 1S

Active?	Yes
Web Page:	http://moodle2425.up.pt/course/view.php?id=6352
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Physics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	9	72	243
L:CC	0	study plan from 2021/22	2	-	9	72	243
			3
L:EF	94	study plan from 2021/22	1	-	9	72	243
L:F	61	Official Study Plan	1	-	9	72	243
			3
L:G	0	study plan from 2017/18	2	-	9	72	243
			3
L:Q	0	study plan from 2016/17	3	-	9	72	243

Teaching language

Portuguese

Objectives

To master the basic concepts, results and techniques of the differential and integral calculus on one variable.

Learning outcomes and competences

Students are expected to master the basic concepts of analysis of real functions in one real variable, namely: sequences, limits, series, continuity,  the derivative, primitive and integral operators, and Taylor polynomials and series.

This course unit is also desinged to provide the adequate environment for students to work rigorously with analytical concepts that previously have only been used at an intuitive level.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

N/A

Program

0. THE SET OF REAL NUMBERS 

The set of real numbers: algebraic structure, ordering and completeness. 

1. LIMITS AND CONTINUITY

Sequences: definition, limit, uniqueness of its limit, monotone and bounded sequences, subsequences. Definition and uniqueness of the limit of a function at a point.  Heine’s characterization of limit. Lateral limits and the arithmetic of limits. Limits at infinity, and horizontal and oblique asymptotes. Infinite limits and vertical asymptotes. Continuous functions. 

2. DERIVATIVES AND ANTIDERIVATIVES

Geometric motivation and physical meaning of the notion of derivative of a real function at a point. Definition of derivative and lateral derivatives at a point. Anti-derivatives. Derivatives and anti-derivatives of elementary functions. Relationship between continuity and derivability. Squeeze theorem. Derivatives and anti-derivatives of sums and products by a scalar. Derivatives of products and quotients. Chain rule and corresponding anti-derivative rule. Derivative of the inverse function. Inverses of trigonometric functions and corresponding derivatives. Anti-derivative by substitution. Anti-derivative by parts. Anti-derivatives of rational functions.

3. INTEGRALS

Concept of area. Riemann integral of a bounded function over an interval. Integrable functions. Basic properties of integrals. The area function. The Fundamental Theorem of Calculus and its consequences. Computation of integrals. Integration by substitution and integration limits. Improper integrals: the case of continuous functions defined on unbounded intervals and the case of continuous unbounded functions defined on an interval.

4. THE FUNDAMENTAL THEOREMS OF CALCULUS AND APPLICATIONS

Theorems of continuity: the permanence of sign in the neighbourhood of a continuity point; Theorem of Intermediate Values; Theorems of Bolzano and Weierstrass.

Theorems on differentiable functions: the derivative is zero in local extreme points (for functions whose domains are open); Mean Value Theorems (Rolle’s, Lagrange’s, Cauchy’s). Applications: determination of extremes of functions; proof that a function defined on an interval with nil derivative is constant; determination of monotonic intervals and concavity; classification of critical points. Indeterminate limits. L’Hôpital’s Rule. Convex functions whose domain is a closed interval. Possible discontinuities of a function which is the derivative of another inside an interval. Functions of class C^k.

5. POLYNOMIAL APPROXIMATION

Polynomial approximation of functions: Taylor polynomials; tangent of degree n of a function and its Taylor polynomial of order n at a given point; Lagrange’s formula for the remainder. Application: irrationality of Neper’s number. Taylor series: convergence domain and error formula.

6 SERIES

Series of real numbers: absolute and conditional convergence; convergence criteria. Series of functions: pointwise and uniform convergence; termwise differentiation and integration. Power series: interval and radius of convergence.

7. COMPLEMENTARY MATERIAL ON INTEGGRALS

Time permitting, further topics such as the characterization of Riemann integrable functions and the gage integral may be considered.

Mandatory literature

Kitchen Jr. Joseph W.; Calculus
Adams Robert A.; Calculus. ISBN: 0-201-39607-6

Complementary Bibliography

Spivak Michael; Calculus. ISBN: 84-291-5139-7 (Vol. 1)
Peter D. Lax , Maria Shea Terrell; Calculus with applications, Springer New York, NY, 2014. ISBN: 978-1-4939-3688-5 (https://link.springer.com/book/10.1007/978-1-4614-7946-8)
Kenneth A. Ross; Elementary analysis. ISBN: 0-387-90459-X (2nd ed. (2013): https://link.springer.com/book/10.1007/978-1-4614-6271-2)

Teaching methods and learning activities

Presentation of the course material by the teacher. Slides presented in the lectures will be made available to the students, as well as exercise sheets. Exercises will be suggested each week for discussion in the TP-classes so as to encourage autonomous work in advance of classes.

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	171,00
Frequência das aulas	72,00
Total:	243,00

Eligibility for exams

No requirements: all registered students are considered to have followed the course.

Calculation formula of final grade

The student assessment in the normal exam period consists in two tests during the semester, each with the weight 0.3, the remaning weight of 0.4 being attributed to the final exam.
Participation in the tests and in the final exam is mandatory; 
students that do not participate in at least one of them fail the course for lack of evaluation component.

The remedial exam has weight 1 in the final grade for students that register for it. Registration for this exam is automatic for students registered for the course that have not previously obtained a passing grade. Previously approved students who wish to improve their grade according to the regulations may also register to take the remedial exam.

Special assessment (TE, DA, ...)

The exams required under the special legal cases will be written, but may be preceded by na oral exam to establish if the student should be admitted to the written exam.

Classification improvement

The general rules apply.

Observations

Article 13 of the General Regulation for Students’ Evaluation in the University of Porto, approved the 19th May 2010: "Any student who commits fraud during an exam or test fails that exam and will face disciplinary charges by the University."